Antigoni Kaliontzopoulou, CIBIO/InBIO, University of Porto
17 October, 2019
With time, these associations lead to the integration among different body parts
Biological systems can also exhibit a modular structure: where some traits are highly correlated with one another, and are less correlated with other sets of structures
Integration describes how characters are correlated with each other
Correlations that are stronger among some subsets of traits than between others (Olson and Miller 1958)
Cohesion among traits that result from interactions of biological processes (Klingenberg 2008)
Traits that are mutually informative of one another, conditional on all other traits under examination (Magwene 2001)
Integration describes how characters are correlated with each other
Correlations that are stronger among some subsets of traits than between others (Olson and Miller 1958)
Cohesion among traits that result from interactions of biological processes (Klingenberg 2008)
Traits that are mutually informative of one another, conditional on all other traits under examination (Magwene 2001)
Methodologically: goal is to identify “exceptional” correlations relative to some model
A complementary concept to integration
Describes sets of characters that exhibit higher correlations among them than they do with other sets:
The concept of integration has been explored in different ways
May sometimes become difficult to grasp
What is the “contrary” of integration? Modularity? Or “disintegration”?
Which is the null hypothesis towards which to test for integration or modularity?
Two complementary questions
Methodologies are all related to correlations within vs. between subsets of traits
Inverse procedures for assessing integration vs. modularity
In both cases different methods are available depending on whether one has an initial idea of the subsets of associated traits
Detect significant correlations, while accounting for correlations with other traits
Procedure
Detect significant correlations, while accounting for correlations with other traits
Procedure
Graphically, this is equivalent to ‘pruning’ links between traits
Method ‘exploratory’ in that modules are not known a priori
In some cases, we have biological hypotheses of putative modules, and wish to know whether they are correlated (integrated) with one another
This test may be accomplished via Multivariate Association measures
Two approaches may be (and have been) used: the RV coefficient and Partial Least Squares
In some cases, we have biological hypotheses of putative modules, and wish to know whether they are correlated (integrated) with one another
This test may be accomplished via Multivariate Association measures
Two approaches may be (and have been) used: the RV coefficient and Partial Least Squares
But first recall (from Covariation lecture):
\(\small\mathbf{S}_{11}\): covariation of variables in \(\small\mathbf{Y}_{1}\)
\(\small\mathbf{S}_{22}\): covariation of variables in \(\small\mathbf{Y}_{2}\)
\(\small\mathbf{S}_{21}=\mathbf{S}_{12}^{T}\): covariation between \(\small\mathbf{Y}_{1}\) and \(\small\mathbf{Y}_{2}\)
\(\small\mathbf{S}_{21}=\mathbf{S}_{12}^{T}\) is the multivariate equivalent of \(\small\sigma_{21}\)
\[\small{RV}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}_{11}\mathbf{S}_{11})tr(\mathbf{S}_{22}\mathbf{S}_{22})}}\] Range of \(\small\mathbf{RV}\): \(\small{0}\rightarrow{1}\)
The RV coefficient is analogous to \(\small{r}^{2}\) but it is not a strict mathematical generalization
\(\small{r}^{2}=\frac{\sigma^{2}_{xy}}{\sigma^{2}_{x}\sigma^{2}_{y}}\) vs. \(\small{RV}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}_{11}\mathbf{S}_{11})tr(\mathbf{S}_{22}\mathbf{S}_{22})}}\)
The numerator of \(\small{r}^{2}\) & \(\small{RV}\) describes the covariation between \(\small\mathbf{Y}_{1}\) & \(\small\mathbf{Y}_{2}\)
The denominator of \(\small{r}^{2}\) & \(\small{RV}\) describes variation within \(\small\mathbf{Y}_{1}\) & \(\small\mathbf{Y}_{2}\)
Thus, \(\small{RV}\) (like \(\small{r}^{2}\)) is a ratio of between-block relative to within-block variation
\[\small{RV}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}_{11}\mathbf{S}_{11})tr(\mathbf{S}_{22}\mathbf{S}_{22})}}\] Range of \(\small\mathbf{RV}\): \(\small{0}\rightarrow{1}\)
The RV coefficient is analogous to \(\small{r}^{2}\) but it is not a strict mathematical generalization
\(\small{r}^{2}=\frac{\sigma^{2}_{xy}}{\sigma^{2}_{x}\sigma^{2}_{y}}\) vs. \(\small{RV}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}_{11}\mathbf{S}_{11})tr(\mathbf{S}_{22}\mathbf{S}_{22})}}\)
The numerator of \(\small{r}^{2}\) & \(\small{RV}\) describes the covariation between \(\small\mathbf{Y}_{1}\) & \(\small\mathbf{Y}_{2}\)
The denominator of \(\small{r}^{2}\) & \(\small{RV}\) describes variation within \(\small\mathbf{Y}_{1}\) & \(\small\mathbf{Y}_{2}\)
Thus, \(\small{RV}\) (like \(\small{r}^{2}\)) is a ratio of between-block relative to within-block variation
However, because each \(\small\mathbf{S}\) is a covariance matrix, the sub-components of \(\small\mathbf{RV}\) are squared variances and covariances: not variances as in \(\small{r}^{2}\): \(\tiny\text{hence, range of } \mathbf{RV}={0}\rightarrow{1}\)
Another way to summarize the covariation between blocks is via Partial Least Squares (PLS)
Decomposing the information in \(\small\mathbf{S}_{12}\) to find rotational solution (direction) that describes greatest covariation between \(\small\mathbf{Y}_{1}\) and \(\small\mathbf{Y}_{2}\)
\[\small\mathbf{S}_{12}=\mathbf{U\Lambda{V}}^T\]
Ordination scores found by projection of centered data on vectors \(\small\mathbf{U}\) and \(\small\mathbf{V}\)
\[\small\mathbf{P}_{1}=\mathbf{Y}_{1}\mathbf{U}\]
\[\small\mathbf{P}_{2}=\mathbf{Y}_{2}\mathbf{V}\]
The first columns of \(\small\mathbf{P}_{1}\) and \(\small\mathbf{P}_{2}\) describe the maximal covariation between \(\small\mathbf{Y}_{1}\) and \(\small\mathbf{Y}_{2}\)
The correlation between \(\small\mathbf{P}_{11}\) and \(\small\mathbf{P}_{21}\) is the PLS-correlation
\[\small{r}_{PLS}={cor}_{P_{11}P_{21}}\]
Significance is assessed via permutation
Pecos pupfish (shape obtained using geometric morphometrics)
Is there an association between head shape and body shape?
Pecos pupfish (shape obtained using geometric morphometrics)
Is there an association between head shape and body shape?
\[\small{RV}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}_{11}\mathbf{S}_{11})tr(\mathbf{S}_{22}\mathbf{S}_{22})}}=0.607\]
\[\small\sqrt{RV}=0.779\]
\(\tiny{RV}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}_{11}\mathbf{S}_{11})tr(\mathbf{S}_{22}\mathbf{S}_{22})}}=0.607\) and \(\tiny\sqrt{RV}=0.779\)
\(\small{r}_{PLS}={cor}_{P_{11}P_{21}}=0.916\)
We now have two potential test measures of multivariate correlation
\[\small{RV}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}_{11}\mathbf{S}_{11})tr(\mathbf{S}_{22}\mathbf{S}_{22})}}\]
\[\small{r}_{PLS}={cor}_{P_{11}P_{21}}\]
Test statistics:
\(\small\hat\rho=\sqrt{RV}\) and \(\small\hat\rho={r}_{PLS}\)
H0: \(\small\rho=0\)
H1: \(\small\rho>0\)
RRPP Approach:
1: Represent \(\small\mathbf{Y}_{1}\) and \(\small\mathbf{Y}_{2}\) as deviations from mean (H0)
2: Estimate \(\small\hat\rho=\sqrt{RV}_{obs}\) and \(\small\hat\rho={r}_{PLS_{obs}}\)
3: Permute rows of \(\small\mathbf{Y}_{2}\), obtain \(\small\hat\rho=\sqrt{RV}_{rand}\) and \(\small\hat\rho={r}_{PLS_{rand}}\)
4: Repeat many times to generate sampling distribution
Test statistics:
\(\small\hat\rho=\sqrt{RV}\) and \(\small\hat\rho={r}_{PLS}\)
H0: \(\small\rho=0\)
H1: \(\small\rho>0\)
For the pupfish dataset, both are significant at p = 0.001
Test statistics:
\(\small\hat\rho=\sqrt{RV}\) and \(\small\hat\rho={r}_{PLS}\)
H0: \(\small\rho=0\)
H1: \(\small\rho>0\)
Compare permutation distributions with one another (minus observed in this case)
plot(RV.rand[-1], pls.rand[-1], xlim=c(0,.65), ylim=c(0,.65), xlab="sqrt(RV)", ylab="r-pls")
abline(a=0,b=1, col="red",lwd=2)How tightly integrated is a structure in an overall sense?
Does not consider subsets of the structure, but rather the whole
How tightly integrated is a structure in an overall sense?
Does not consider subsets of the structure, but rather the whole
High integration is expressed as high global correlation among traits
The scaled variance of eigenvalues (SVE) can be used as a measure of global integration
A major, yet previously unaddressed question concerning global integration was: “Integrated” as opposed to what?
That is, what is the null model against which the pattern is being compared?
All previous approaches consider a lack of correlation to be the null, which is a poor null model
For variance in non-uniform shape variation versus variance in BE, the expected value of this relationship is: \(\small\beta= -1\)
For this value, shape patterns are maintained across spatial scales
This is called self-similarity and it can be used as the \(H_0\) to examine the degree of integration of biological data
Deviations in slope from this value provide useful heuristics regarding global integration
## BEval
## -0.6369197
Modularity addresses a question complementary to that of integration
Modules: tightly integrated sets of traits, which are relatively independent from other such sets
In terms of covariation patterns: look for subunits of characters with high within-unit covariation, and little covariation with other subunits
We can identify modules by examining the covariance structure of the data
Divide structure to subunits and calculate the covariation between them
One could consider using the \(\small{RV}\) coefficient to evaluate modularity, where small \(\small{RV}\) values signify greater modularity
However, the \(\small{RV}\) coeffiicent varies systematically with both n and p, meaning the ‘expected’ value representing modularity is arbitrary!
With random MVN data, \(\small\mathbf{RV}\) varies with both n and p!
\[\small{CR}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}^*_{11}\mathbf{S}^*_{11})tr(\mathbf{S}^*_{22}\mathbf{S}^*_{22})}}\]
\[\small{CR}=\frac{tr(\mathbf{S}_{12}\mathbf{S}_{21})}{\sqrt{tr(\mathbf{S}^*_{11}\mathbf{S}^*_{11})tr(\mathbf{S}^*_{22}\mathbf{S}^*_{22})}}\]
where \(\small\mathbf{S}^*_{11}\) & \(\small\mathbf{S}^*_{22}\) represent the within-module covariance matrices with \(\small{0}\) along the diagonal
The \(\small{CR}\) coefficient does NOT vary with n and p:
It may be of interest to know whether the ‘degree’ of integration (or modularity) is greater in one dataset as compared to another
How can one evaluate such patterns?
It may be of interest to know whether the ‘degree’ of integration (or modularity) is greater in one dataset as compared to another
How can one evaluate such patterns?
One cannot directly compare \(\small{RV}\) or \(\small{r}_{PLS}\) patterns because both vary with n and p
where: \(\small\mathbf{Z}=\frac{r_{PLS_{obs}}-\mu_{r_{PLS_{rand}}}}{\sigma_{r_{PLS_{rand}}}}\)
Integration and modularity are of relevance for many E&E questions
Several different aspects can be explored: global integration vs. integration of different structures; with vs. without a priori hypotheses etc.
Careful with definitions and matching between concepts and stats
RV has been extensively used to assess modularity, but it exhibits undesirable statistical properties
CR resolves these and can be promptly used to test for modularity
Open questions regarding integration
Generally, indices used depend on N and p (at least), so further modifications may be required